Qubit-optical-CMOS integration using structured substrates

ABSTRACT

Techniques for the integration of SiGe/Si optical resonators with qubit and CMOS devices using structured substrates are provided. In one aspect, a waveguide structure includes: a wafer; and a waveguide disposed on the wafer, the waveguide having a SiGe core surrounded by Si, wherein the wafer has a lower refractive index than the Si (e.g., sapphire, diamond, SiC, and/or GaN). A computing device and a method for quantum computing are also provided.

FIELD OF THE INVENTION

The present invention relates to optical resonators, and moreparticularly, to the integration of silicon germanium (SiGe)/silicon(Si) optical resonators with quantum bit (i.e., “qubit”) andcomplementary metal oxide semiconductor (CMOS) devices using structuredsubstrates.

BACKGROUND OF THE INVENTION

The ability to link microwave electrical signals and optical photons forquantum information processing requires efficient conversion between themicrowave and optical domains. See, for example, Rueda et al.,“Efficient microwave to optical photon conversion: an electro-opticalrealization,” Optica, vol. 3, no. 6, pp. 597-604 (June 2016).

A microwave-to-optical transducer can be employed for this conversion.See, for example, U.S. Pat. No. 9,857,609 issued to Bishop et al.,entitled “Integrated Microwave-to-Optical Single-Photon Transducer withStrain-Induced Electro-Optic Material” (hereinafter “U.S. Pat. No.9,857,609”) which describes a quantum computing device having atransducer for converting single-photon microwave signals to opticalsignals. U.S. Pat. No. 9,857,609 provides an integrated optical designincorporating qubits, a transducer and an optical resonator that operatein conjunction to convert microwave signals from the qubits to opticalsignals. The optical signals couple with a waveguide and are transmittedto their destination. However, by employing a silicon germanium (SiGe)on silicon (Si) substrate architecture, optical leakage can be aconcern.

Accordingly, improved designs for integrating quantum computing andoptical devices on a common substrate wafer would be desirable.

SUMMARY OF THE INVENTION

The present invention provides techniques for the integration of SiGe/Sioptical resonators with qubit and CMOS devices using structuredsubstrates. In one aspect of the invention, a waveguide structure isprovided. The waveguide structure includes: a wafer; and a waveguidedisposed on the wafer, the waveguide having a SiGe core surrounded bySi, wherein the wafer has a lower refractive index than the Si. Forexample, the wafer can include a material such as sapphire, diamond,silicon carbide (SiC), and/or gallium nitride (GaN).

In another aspect of the invention, a computing device is provided. Thecomputing device includes: a waveguide structure having a wafer, and awaveguide disposed on the wafer, the waveguide being a resonator-basedmicrowave-to-optical transducer having a SiGe core surrounded by Si,wherein the wafer has a lower refractive index than the Si; qubitsdisposed on the wafer; superconducting bus paths between the waveguideand the qubits; and FETs disposed on the wafer connecting thesuperconducting bus paths between the waveguide and the qubits.

In yet another aspect of the invention, a method for quantum computingis provided. The method includes: providing a computing device thatincludes: (i) a waveguide structure having a wafer, and a waveguidedisposed on the wafer, the waveguide being a resonator-basedmicrowave-to-optical transducer having a SiGe core surrounded by Si,wherein the wafer has a lower refractive index than the Si, (ii) qubitsdisposed on the wafer, (iii) superconducting bus paths between thewaveguide and the qubits, and (iv) FETs disposed on the wafer connectingthe superconducting bus paths between the waveguide and the qubits;selecting one of the superconducting bus paths between the waveguide anda given one of the qubits; routing a microwave signal from the given oneof the qubits along the one of the superconducting bus paths that hasbeen selected; and converting the microwave signal to an optical signalvia the resonator-based microwave-to-optical transducer.

A more complete understanding of the present invention, as well asfurther features and advantages of the present invention, will beobtained by reference to the following detailed description anddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional diagram illustrating an exemplary waveguidestructure according to an embodiment of the present invention;

FIG. 2 is a diagram illustrating an exemplary methodology for formingthe present waveguide structure according to an embodiment of thepresent invention;

FIG. 3 is a top-down diagram illustrating the present waveguidestructure forming a closed loop optical ring resonator-basedmicrowave-to-optical transducer according to an embodiment of thepresent invention;

FIG. 4 is a cross-sectional diagram illustrating qubits, photonics androuting field-effect transistors (FETs) integrated on a common lowerrefractive index wafer according to an embodiment of the presentinvention;

FIG. 5 is a top-down diagram illustrating an exemplary quantum computingdevice containing qubits, photonics and FET components integrated on acommon lower refractive index wafer according to an embodiment of thepresent invention; and

FIG. 6 is a diagram illustrating an exemplary methodology for quantumcomputing according to an embodiment of the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Provided herein are techniques for the integration of silicongermanium/silicon (SiGe/Si) optical resonators/transducers with quantumbit (qubit) and complementary metal oxide semiconductor (CMOS) devicesusing structured substrates (e.g., substrates formed from multiplematerials). The structured substrates have a silicon-on-X (SOX)configuration, where X is a wafer having a material with i) a lowerrefractive index than Si and ii) a radio frequency (RF) loss tangent atmicrowave frequencies of <1×10⁻⁵ at 10 mK for the qubit operation. Byway of example only, suitable materials X that can meet thesequalifications include, but are not limited to, sapphire, diamond,silicon carbide (SiC) and/or gallium nitride (GaN).

A unique SiGe/Si optical resonator is provided whereby this structuredSOX substrate provides a secondary, higher index contrast interface tothe SiGe/Si waveguide definition. The present structured substrate-basedoptical resonator designs have the following notable benefits. First,the thickness and germanium (Ge) mole fraction of the SiGe layers can bereduced while achieving the same radiation limited quality factor due tothe refractive index contrast the lower refractive index X wafer, e.g.,sapphire, diamond, SiC and/or GaN, provides. The refractive indexcontrast provided by the lower refractive index X wafer reduces thespatial extent of the evanescent optical mode and therefore enables thecloser placement of a bottom electrode to the waveguide, which enhancesperformance by increasing the single photon field strength for quantumapplications.

As will be described in detail below, according to an exemplaryembodiment, the present SiGe/Si optical resonators are integrated withqubit and CMOS devices (e.g., transistors) on the same SOX structuredsubstrate. These devices can be connected together to have newfunctionality. Further, techniques are presented herein for introducingreconfiguration capability in networks of qubits and microwave-opticaltransducers using reconfigurable electrical routing by the transistors.

FIG. 1 is a diagram illustrating an exemplary waveguide structure inaccordance with the present techniques. As shown in FIG. 1 , waveguidestructure includes a waveguide 100 disposed on a low refractive indexwafer 104, i.e., wafer 104 has a lower refractive index than Si layer102. As provided above, suitable materials for wafer 104 include, butare not limited to, sapphire, diamond, SiC and/or GaN. The lower indexwafer under the waveguide 100 serves to truncate the evanescent opticalfield (for example allowing the placement of a bottom electrode 106closer to the waveguide 100) and minimize radiation loss. As will bedescribed in detail below, fabrication of the waveguide structure (andoptional associated computing components such as qubits and/or routingtransistors) can begin with an SOX wafer, i.e., Si layer 102 on Xmaterial (e.g., sapphire, diamond, SiC and/or GaN) wafer 104.

Specifically, waveguide 100 includes a SiGe core 108 surrounded by Silayer 102 and an Si layer 110. SiGe core 108 has a refractive index thatis higher than the refractive index of the Si layers 102/110. With thismaterial combination and configuration, the waveguide 100 providesprimary modal index guiding.

The present waveguide 100 can have a variety of different applications.For instance, the present waveguide 100 can be employed as an opticalrouting waveguide for transmitting optical signals and/or as an opticalring resonator filter. For an optical ring resonator filter, waveguide100 is configured as a closed loop optical ring resonator which acts asa filter for light of a certain (resonant) wavelength.

According to an exemplary embodiment of the present techniques,waveguide 100 is employed as a microwave-to-optical transducer thatconverts signals from qubits at microwave frequencies into opticalsignals at a telecommunication frequency (e.g., about 1550 nanometers(nm) wavelength). See, for example, exemplary computing device 500,described in conjunction with the description of FIG. 5 , below. Asdescribed in detail below, multiple physical superconducting bus pathscan be present between the qubits and the transducer with multipletransistors connecting the paths. The transistors provide forreconfigurable connections (electrical routing) to the transducer.Advantageously, the qubits, CMOS transistors, and transducer are allintegrated on a single common SOX substrate.

For use as a microwave-to-optical transducer, waveguide 100 preferablyincludes a bottom electrode 106, a top electrode 112, and biaselectrodes 114. As shown in FIG. 1 , bottom electrode 106 is disposed ona side of the wafer 104 opposite the waveguide 100, top electrode 112 isdisposed on the waveguide 100 (i.e., on top of the Si layer 110), andthe bias electrodes 114 are disposed on the wafer 104 on opposite sidesof the waveguide 100. These electrodes form a transmission lineresonator in which there is a standing wave radio frequency (RF) field.Microwave frequencies generally refer to those RF frequencies of greaterthan or equal to about 1 gigahertz (GHz). Thus, the transmission lineresonator spatially overlaps the RF and optical modes. According to anexemplary embodiment, the Kerr electro-optic effect or DC Kerr effect isused for microwave-to-optical conversion. As known in the art, the DCKerr effect refers to a change in the refractive index of a material inresponse to a slowly varying electrical field applied to the samplematerial. Here, the electric field in applied via the bias electrodes114 or by backgating the waveguide 100 via the bottom electrode 106through the wafer 104. As provided above, the refractive index contrastprovided by the lower refractive index X wafer 104 reduces the spatialextent of the evanescent optical mode which enables the closer placementof the bottom electrode 106 to the waveguide 100, and thus enhancesperformance by increasing the single photon field strength. As will bedescribed in conjunction with the description of FIG. 2 below, the topelectrode 112 can be configured as a crescent shape over the opticalring resonator forming a quarter-wave RF resonator which provides closeintegration of the electrode with the optical ring resonator.

Standard CMOS-compatible fabrication process can be employed to form thewaveguide structure. See, for example, methodology 200 of FIG. 2 . Byway of example only, the process begins in step 202 with a SOX substratehaving an SOX layer 102 (a Si layer) on lower refractive index X wafer104 (e.g., sapphire, diamond, SiC and/or GaN).

In step 204, standard lithography and etching techniques are thenemployed to pattern the SOX layer 102 into the respective shape ofwaveguide structure using, e.g., a directional (anisotropic) etchingprocess such as reactive ion etching (RIE). For instance, the SOX layer102 can be patterned into an optical routing waveguide of any shape. Forinstance, for use as an optical ring resonator filter or optical ringresonator-based microwave-to-optical transducer, the SOX layer 102 ispatterned into a closed loop/ring.

In step 206, SiGe is disposed (i.e., deposited, grown, etc.) on thepatterned SOX layer 102. SiGe has a refractive index that is higher thanthe refractive index of Si. In step 208, standard lithography andetching techniques are then employed to pattern the SiGe the shape ofcore 108 using, e.g., a directional (anisotropic) etching process suchas RIE. The patterned core 108 will have the same general shape as thepatterned SOX layer 102.

Finally, in step 210, the core 108 is covered in an Si layer 110 tocomplete the waveguide structure 100. Again, standard lithography andetching techniques can be employed to pattern the Si layer 110 into therespective shape of the waveguide 100 surrounding the core 108.

FIG. 3 is a top-down diagram illustrating an exemplary embodimentwherein the waveguide 100 forms a closed loop optical ringresonator-based microwave-to-optical transducer. As is visible from thetop-down view in FIG. 3 , the waveguide 100 is an optical ring resonatorformed on lower refractive index X wafer 104 (e.g., sapphire, diamond,SiC and/or GaN). The top electrode 112 is configured as a crescent overthe waveguide 100 forming a quarter-waver RF resonator (see above). Forclarity, the bias electrode and/or ground plane is omitted from thisdrawing.

The waveguide 100 is optically coupled to a routing waveguide bus 302.Routing waveguide bus 302 transfers optical signals waveguide 100 totheir destination. According to an exemplary embodiment, the routingwaveguide bus 302 is formed on wafer 104 in the same manner as waveguide100 and includes the same components, e.g., SOX layer/Si layer 110surrounding a (e.g., SiGe) core 108. However, routing waveguide bus 302would not need the electrodes 106/112/114 associated with thetransducer.

A microwave waveguide 304 is also optically coupled to the waveguide100. In general, the microwave waveguide 304 can be any waveguide busfor carrying microwave photons. For instance, according to an exemplaryembodiment, the microwave waveguide 304 is formed on wafer 104 in thesame manner as waveguide 100 and includes the same components, e.g., SOXlayer/Si layer 110 surrounding a (e.g., SiGe) core 108. As will bedescribed in detail below, embodiments are provided herein wherewaveguide 100 is integrated on wafer 104 with other quantum computingdevices, such as qubits and optionally transistors for reconfigurableconnections (electrical routing) from the qubits to the transducer. Inthat case, microwave waveguide 304 can be employed as a signal bus tothe transducer. See below.

For instance, FIG. 4 is a cross-sectional diagram illustrating qubits,photonics and routing field-effect transistors (FET) integrated on acommon lower refractive index X wafer 104 (e.g., sapphire, diamond, SiCand/or GaN). An exemplary layout of these devices is described inconjunction with the description of FIG. 5 , below. First, regarding thephotonics component, as shown in FIG. 4 the integrated design includesat least one waveguide 100. Namely, as provided above, the presentwaveguide 100 can be configured as an optical routing waveguide(s)and/or an optical ring resonator filter, as well as an optical ringresonator-based microwave-to-optical transducer. Thus, one or more ofthese devices can be incorporated into the present design. As shown inFIG. 5 , and as described above, each waveguide 100 includes, e.g., SOXlayer/Si layer 110 surrounding a SiGe core 108.

At least one qubit is formed on the common lower refractive index Xwafer 104. For instance, as shown in FIG. 4 , each qubit includes aqubit circuit 402 formed on the SOX layer 102. Optionally, a region ofthe SOX layer 102 is undercut beneath the qubit circuit 402 such thatthe qubit circuit is suspended over the wafer 104.

According to an exemplary embodiment, the qubit circuit 402 is aJosephson tunnel junction formed by two superconducting thin films(i.e., a superconducting bottom electrode 406 and a superconducting topelectrode 408) separated by an insulator 410. The Josephson tunneljunction is described generally in Devoret et al., “SuperconductingQubits: A Short Review,” arXiv:cond-mat/0411174 (February 2008) (41pages), the contents of which are incorporated by reference as if fullyset forth herein. Suitable materials for the superconducting bottom/topelectrodes 406/408 include, but are not limited to, aluminum (Al) and/ortitanium nitride. Suitable materials for the insulator 410 include, butare not limited to, oxides such as aluminum oxide (Al₂O₃). For instance,according to an exemplary embodiment, superconducting bottom electrode406/insulator 410/superconducting top electrode 408 are Al/Al₂O₃/Al,respectively.

Standard CMOS-compatible processes may be employed to fabricate thequbit(s) on the wafer 104. Qubit fabrication techniques that may beemployed in accordance with the present techniques are described, forexample, in U.S. Patent Application Publication Number 2015/0340584 byChang et al., entitled “Suspended Superconducting Qubits” (hereinafter“U.S. Patent Application Publication Number 2015/0340584”), the contentsof which are incorporated by reference as if fully set forth herein.

Optionally, a region of the SOX layer 102 is undercut beneath the bottomelectrode 406 such that the qubit is suspended over the wafer 104. Byway of example only, the SOX layer 102 can be undercut using a zenondifluoride (XeF₂) etch. Undercutting the SOX layer 102 is undercutbeneath the qubit circuit 402 is beneficial as it removes thesubstrate-to-metal interface and moves the substrate-to-metal interfacefurther away from the electric fields of the resonant modes of thequantum circuit. See U.S. Patent Application Publication Number2015/0340584.

According to an exemplary embodiment, at least one FET 412 is formed onthe common lower refractive index X wafer 104. See FIG. 4 . In general,each FET 412 includes a channel interconnecting a source and a drain,and a gate for regulating current flow through the channel. In thisexemplary embodiment, the channel is formed from the SOX layer 102. Anystandard FET designs including, but not limited to, planar or non-planarFET configurations such as finFETs, nanowire/nanosheet FETs, etc. may beimplemented in accordance with the present techniques. The techniquesfor forming these planar and non-planar FET designs are well known inthe art.

Source and drains 413 and 414 are formed on opposite ends of thechannel/SOX layer 102. According to an exemplary embodiment, the sourceand drains 413 and 414 are formed from an in-situ or ex-situ dopedepitaxial material such as phosphorous-doped silicon (Si:P) orboron-doped SiGe (SiGe:B). Contacts 416 and 418 are formed to the sourceand drains 413 and 414, respectively.

A top gate 420 and/or a bottom gate 422 can be implemented to regulatecurrent flow through the channel. Each gate generally includes aconductor or combination of conductors separated from the channel by agate dielectric. Suitable gate conductors include, but are not limitedto, doped poly-silicon and/or metals such as titanium (Ti), tantalum(Ta), titanium nitride (TiN), tantalum nitride (TaN), and/or tungsten(W). Suitable gate dielectrics include, but are not limited to, siliconoxide (SiOx) and/or high-x dielectrics such as hafnium oxide (HfO₂)and/or lanthanum oxide (LaO₂).

An exemplary quantum computing device 500 containing qubits, photonicsand FET components integrated on a common lower refractive index X wafer104 is shown in FIG. 5 (a top-down view). Specifically, as shown in FIG.5 , device 500 includes qubits 402 interconnected via multiple physicalsuperconducting bus paths 502 a,b,c,d, etc. to waveguide 100 which isconfigured as an optical ring microwave-to-optical transducer. MultipleFETs 412 connect the superconducting bus paths 502 a,b,c,d, etc. betweenthe waveguide 100 and the qubits 402. Each of these device components,which was described in detail above, is disposed on wafer 104.

FETs 412 enable reconfigurable electrical routing between the waveguide100 and the qubits 402 allowing the devices to be connected together tohave new functionality. For instance, each superconducting bus paths 502a,b,c,d, etc. is connected to a separate FET 412. See FIG. 5 . Namely,each superconducting bus paths 502 a,b,c,d, etc. is connected to thesource and drains (i.e., source and drains 413 and 414—see FIG. 4 ) ofthe respective FET 412. A contact 504 is provided to the gate (i.e., topgate 420 and/or a bottom gate 422—see FIG. 4 ) of each FET 412. Applyinga gate voltage (via contact 504) to a FET 412 switches the respectiveFET 412 from a high resistance state between the source and drains ofthat FET 412 to a low resistance state between the source and drains.The superconducting bus paths 502 a,b,c,d, etc. with the low resistanceswitch is then where the microwave signal will be routed. See, forexample, methodology 600 of FIG. 6 .

Namely, FIG. 6 provides an exemplary methodology 600 for quantumcomputing using a device such as quantum computing device 500 of FIG. 5. As provided above, computing device 500 includes qubits 402interconnected via multiple physical superconducting bus paths 502a,b,c,d, etc. to optical ring microwave-to-optical transducer waveguide100. Multiple FETs 412 connect the superconducting bus paths 502a,b,c,d, etc. between the waveguide 100 and the qubits 402.

In step 602, a gate voltage is applied to at least one of the FETs 412to select at least one of the superconducting bus paths 502 a,b,c,d,etc. Namely, as provided above, each superconducting bus paths 502a,b,c,d, etc. is connected to the source and drains (i.e., source anddrains 413 and 414—see FIG. 4 ) of the respective FET 412, and applyinga gate voltage (via contact 504) to a FET 412 switches the respectiveFET 412 from a high resistance state between the source and drains ofthat FET 412 to a low resistance state between the source and drains.

In step 604, a microwave signal is routed from the qubit 402 along theselected superconducting bus path 502 a,b,c,d, etc. to the optical ringmicrowave-to-optical transducer waveguide 100. Namely, as providedabove, the superconducting bus paths 502 a,b,c,d, etc. with the lowresistance switch is where the microwave signal will be routed. In step606, the optical ring microwave-to-optical transducer waveguide 100converts the microwave signal to an optical signal.

Although illustrative embodiments of the present invention have beendescribed herein, it is to be understood that the invention is notlimited to those precise embodiments, and that various other changes andmodifications may be made by one skilled in the art without departingfrom the scope of the invention.

What is claimed is:
 1. A method for quantum computing, the methodcomprising: providing a computing device that includes: (i) a waveguidestructure comprising a wafer, and a waveguide disposed on the wafer, thewaveguide comprising a resonator-based microwave-to-optical transducerhaving a silicon germanium (SiGe) core surrounded by silicon (Si),wherein the wafer has a lower refractive index than the Si, (ii) quantumbits (qubits) disposed on the wafer, (iii) superconducting bus pathsbetween the waveguide and the qubits, and (iv) field effect transistors(FETs) disposed on the wafer connecting the superconducting bus pathsbetween the waveguide and the qubits; selecting one of thesuperconducting bus paths between the waveguide and a given one of thequbits; routing a microwave signal from the given one of the qubitsalong the one of the superconducting bus paths that has been selected;and converting the microwave signal to an optical signal via theresonator-based microwave-to-optical transducer.
 2. The method of claim1, wherein the wafer comprises a material selected from a groupconsisting of: sapphire, diamond, silicon carbide (SiC), gallium nitride(GaN), and combinations thereof.
 3. The method of claim 1, wherein thesuperconducting bus paths are connected to sources and drains of theFETs.
 4. The method of claim 3, wherein the selecting of the one of thesuperconducting bus paths comprises: applying a gate voltage to a givenone of the FETs along the one of the superconducting bus paths to switchthe given one of the FETs from a high resistance state between thesources and drains to a low resistance state between the sources anddrains.
 5. The method of claim 1, wherein the waveguide has a ringshape.
 6. The method of claim 1, wherein the waveguide structure furthercomprises a top electrode disposed on the waveguide.
 7. The method ofclaim 6, wherein the top electrode has a crescent shape.
 8. The methodof claim 1, wherein the waveguide structure further comprises a bottomelectrode disposed on a side of the wafer opposite the waveguide.
 9. Themethod of claim 1, wherein the waveguide structure further comprisesbias electrodes disposed on the wafer on opposite sides of thewaveguide.
 10. The method of claim 1, wherein each of the qubitscomprises a superconducting bottom electrode on a Si layer; and asuperconducting top electrode separated from the superconducting bottomelectrode by an insulator.
 11. The method of claim 10, wherein the Silayer is undercut beneath the superconducting bottom electrode such thateach of the qubits is suspended over the wafer.
 12. A method for quantumcomputing, the method comprising: providing a computing device thatincludes: (i) a waveguide structure comprising a wafer, and a waveguidedisposed on the wafer, the waveguide comprising a resonator-basedmicrowave-to-optical transducer having a SiGe core surrounded by Si,wherein the wafer has a lower refractive index than the Si, (ii) quantumbits (qubits) disposed on the wafer, (iii) superconducting bus pathsbetween the waveguide and the qubits, and (iv) field effect transistors(FETs) disposed on the wafer connecting the superconducting bus pathsbetween the waveguide and the qubits, wherein the superconducting buspaths are connected to sources and drains of the FETs, and wherein eachof the FETs comprises a top gate or a bottom gate to regulate currentflow through a channel between the sources and drains, and a contact tothe top gate or the bottom gate; selecting one of the superconductingbus paths between the waveguide and a given one of the qubits byapplying a gate voltage, via the contact, to a given one of the FETsalong the one of the superconducting bus paths to switch the given oneof the FETs from a high resistance state between the sources and drainsto a low resistance state between the sources and drains; routing amicrowave signal from the given one of the qubits along the one of thesuperconducting bus paths that has been selected; and converting themicrowave signal to an optical signal via the resonator-basedmicrowave-to-optical transducer.
 13. The method of claim 12, wherein thewafer comprises a material selected from a group consisting of:sapphire, diamond, SiC, GaN, and combinations thereof.
 14. The method ofclaim 12, wherein the waveguide has a ring shape.
 15. The method ofclaim 12, wherein the waveguide structure further comprises a topelectrode disposed on the waveguide.
 16. The method of claim 15, whereinthe top electrode has a crescent shape.
 17. The method of claim 12,wherein the waveguide structure further comprises a bottom electrodedisposed on a side of the wafer opposite the waveguide.
 18. The methodof claim 12, wherein the waveguide structure further comprises biaselectrodes disposed on the wafer on opposite sides of the waveguide. 19.The method of claim 12, wherein each of the qubits comprises asuperconducting bottom electrode on a Si layer; and a superconductingtop electrode separated from the superconducting bottom electrode by aninsulator.
 20. The method of claim 19, wherein the Si layer is undercutbeneath the superconducting bottom electrode such that each of thequbits is suspended over the wafer.